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On four-point connectivities in the critical 2d Potts model

by Marco Picco, Sylvain Ribault, Raoul Santachiara

This is not the current version.

Submission summary

As Contributors: Sylvain Ribault
Arxiv Link:
Date submitted: 2019-06-25
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics
Discipline: Physics
Subject area: High-Energy Physics - Theory
Approaches: Theoretical, Computational


We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states $Q\in (0,4)$ that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the $2$ to $4$ significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases $Q=0, 3, 4$. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane $\{\Re c <13\}$, where $c$ is the central charge of the Virasoro symmetry algebra.

Ontology / Topics

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Conformal field theory (CFT) Monte-Carlo simulations Potts model Virasoro algebras
Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2019-7-30 Invited Report

  • Cite as: Anonymous, Report on arXiv:1906.02566v1, delivered 2019-07-30, doi: 10.21468/SciPost.Report.1082


The paper is devoted to four-point connectivities in the 2D Q-state Potts model at criticality. The problem of connectivities - probabilities for the distribution of N points among clusters - is fundamental for the theory of geometrical critical phenomena such as percolation (Q=1), but exact results have been missing for long time in the case of points in the bulk. New interest developed in the last years after that Delfino and Viti obtained the result for N=3 and studied the general relation with spin correlators, in particular for N=4. As part of these developments, the authors of the present paper mumerically implemented a conformal bootstrap for N=4 based on a conjecture for the spectrum of operators appearing in the operator product expansion [7]. The conjecture was supported by the numerical consistency of the bootstrap and by comparison with Monte Carlo simulations. More recently, however, Jacobsen and Saleur managed to show by a transfer matrix study that the spectrum conjectured in [7] is incomplete and proposed how to extend it. In the present paper the authors further compare Monte Carlo results for the N=4 Potts connectivities with their previous conjecture, which corresponds to a conformal field theory that they call the "odd CFT". They show that the latter provides results that agree with Monte Carlo data within the accuracy achievable in their simulations, but at the same time they confirm by analytic arguments that the four-point correlators of the odd CFT cannot be the exact answer for the Potts connectivities for Q generic, in particular for Ising (Q=2) and percolation. On the other hand, they argue that the answer is exact for Q=0,3,4, and show analytic coincidence with independent calculations for these values.

The paper is interesting and provides a further step in the study of an important and difficult problem. I have only few minor comments:

1. The authors write at the beginning of section 3.3 that they will compare correlators of the odd CFT to Potts connectivities, and that they will also give an exact linear relation between them. I understand that they refer to Eq. (3.33), which however is not exact, as they stress after Figure 3.1. This point can be made more clear.

2. The expansion (3.36) contains P_\sigma on both sides and can be written more clearly.

3. The basic relation (1.2) calls for a reference (Dotsenko, Fateev, 1984).

4. Eq. (2.16) extends results obtained for Liouville theory and its imaginary version, in particular by Zamolodchikov and Zamolodchikov. I suggest a reference to that work.

5. Ref. [18] is at the origin of the developments of the last years on CFT of connectivities and should be quoted in the introduction.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2019-7-26 Invited Report

  • Cite as: Anonymous, Report on arXiv:1906.02566v1, delivered 2019-07-26, doi: 10.21468/SciPost.Report.1074


The authors extend their previous work and compare data for four-point cluster connectivities obtained in Monte-Carlo simulations with CFT predictions. These computations are done for the two-dimensional Potts model. Whereas very good numerical agreement is found with the predictions from the odd CFT, they provide arguments that this agreement is not exact for general values of Q.

This is an interesting paper that adds to the discussion of solving the Potts model for arbitrary values of Q. Based on the content I can recommend its publication. However, I do object to the format of the paper that is neither traditional, nor is it in agreement with the SciPost manuscript preparation guidelines. The authors made the choice to not provide all the necessary information to identify a reference, but instead mainly give the authors' names and the titles (and sometimes the arXiv number). References should allow the reader to quickly locate a relevant paper, but this is not possible without journal name, volume, first page (or paper identifier). In the first half of the paper figures and tables are not labeled nor are they accompanied by a descriptive caption. Figures in the second half of the paper (that present the numerical data) systematically do not have axes labels. The paper ends without a summary or discussion of the main results. The authors should adhere to accepted standards for a scientific publication before this manuscript is accepted.

Overall, the paper is written in good English, but there are still some typos (like 'phenomenons' and 'a four-point functions encodes" on page 2) that need to be corrected.

  • validity: high
  • significance: good
  • originality: good
  • clarity: good
  • formatting: below threshold
  • grammar: good

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